The *expected utility hypothesis* stems from Daniel Bernoulli's (1738) solution to the famous St.
Petersburg Paradox posed in 1713 by his cousin Nicholas Bernoulli (it is common to note
that Gabriel Cramer, another Swiss mathematician, also provided effectively the same
solution ten years before Bernoulli). The Paradox challenges the old idea that people
value random ventures according to its expected return. The Paradox posed the following
situation: a fair coin will be tossed until a head appears; if the first head appears on
the nth toss, then the payoff is 2^{n} ducats. How much should one pay to play
this game? The paradox, of course, is that the expected return is infinite, namely:

E(w) = ・/font>

_{i=1･ }(1/2^{n})ｷ2^{n}= (1/2)ｷ2 + (1/4)2^{2}+ (1/8)2^{3}+ .... = 1 + 1 + 1 + ..... = ･

Yet while the expected payoff is infinite, one would not suppose, at least intuitively, that real-world people would be willing to pay an infinite amount of money to play this!

Daniel Bernoulli's solution involved two
ideas that have since revolutionized economics: firstly, that people's utility from
wealth, u(w), is not linearly related to wealth (w) but rather increases at a decreasing
rate - the famous idea of *diminishing marginal utility*, u｢
(Y) > 0 and u｢ ｢ (Y) < 0;
(ii) that a person's valuation of a risky venture is not the expected return of that
venture, but rather the *expected utility* from that venture. In the St. Petersburg
case, the *value* of the game to an agent (assuming initial wealth is zero) is:

E(u) = ・/font>

_{i=1･ }(1/2^{n})ｷu(2^{n}) = (1/2)ｷu(2) + (1/4)ｷu(2^{2}) + (1/8)ｷu(2^{3}) + .... < ･

which Bernoulli conjectured is finite because of the principle of diminishing marginal
utility. (Bernoulli originally used a logarithmic function of the type u(x) = a log x). Consequently, people would only be willing to pay a *finite*
amount of money to play this, even though its expected return is infinite. In general, by
Bernoulli's logic, the valuation of any risky venture takes the expected utility form:

E(u | p, X) = ・/font>

_{xﾎ X}p(x)u(x)

where X is the set of possible outcomes, p(x) is the probability of a particular outcome x ﾎ X and u: X ｮ R is a utility function over outcomes.

[Note: as Karl Menger (1934) later
pointed out, placing an ironical twist on all this, Bernoulli's hypothesis of diminishing
marginal utility is actually not enough to solve all St. Petersburg-type Paradoxes. To see
this, note that we can always find a sequence of payoffs x_{1}, x_{2}, x_{3},
.., which yield infinite expected value, and then propose, say, that u(x_{n}) = 2^{n}
so that expected utility is also infinite. Thus, Menger proposed that utility must also be
bounded above for paradoxes of this type to be resolved.]

Channelled by Gossen (1854), Bernoulli's idea
of diminishing marginal utility of wealth became a centerpiece in the Marginalist Revolution of 1871-4 in the work of Jevons (1871), Menger
(1871) and Walras (1874). However, Bernoulli's
expected utility hypothesis has a thornier history. With only a handful of exceptions
(e.g. Marshall, 1890: pp.111-2, 693-4; Edgeworth, 1911), it was never really picked up until
John von Neumann and Oskar Morgenstern's (1944) *Theory of Games and
Economic Behavior*, which we turn to next.

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